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dct.m

## Copyright (C) 2001 Paul Kienzle
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program; if not, write to the Free Software
## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA

## y = dct (x, n)
##    Computes the disrete cosine transform of x.  If n is given, then
##    x is padded or trimmed to length n before computing the transform.
##    If x is a matrix, compute the transform along the columns of the
##    the matrix. The transform is faster if x is real-valued and even
##    length.
##
## The discrete cosine transform X of x can be defined as follows:
##
##               N-1
##   X[k] = w(k) sum x[n] cos (pi (2n-1) k / 2N ),  k = 0, ..., N-1
##               n=0
##
## with w(0) = sqrt(1/N) and w(k) = sqrt(2/N), k = 1, ..., N-1.  There
## are other definitions with different scaling of X[k], but this form
## is common in image processing.
##
## See also: idct, dct2, idct2, dctmtx

## From Discrete Cosine Transform notes by Brian Evans at UT Austin,
## http://www.ece.utexas.edu/~bevans/courses/ee381k/lectures/09_DCT/lecture9/
## the discrete cosine transform of x at k is as follows:
##
##          N-1
##   X[k] = sum 2 x[n] cos (pi (2n-1) k / 2N )
##          n=0
##
## which can be computed using:
##
##   y = [ x ; flipud (x) ]
##   Y = fft(y)
##   X = exp( -j pi [0:N-1] / 2N ) .* Y
##
## or for real, even length x
##
##   y = [ even(x) ; flipud(odd(x)) ]
##   Y = fft(y)
##   X = 2 real { exp( -j pi [0:N-1] / 2N ) .* Y }
##
## Scaling the result by w(k)/2 will give us the desired output.

## Author: Paul Kienzle
## 2001-02-08
##   * initial release
function y = dct (x, n)

  if (nargin < 1 || nargin > 2)
    usage ("y = dct(x [, n])");
  endif

  realx = isreal(x);
  transpose = (rows (x) == 1);

  if transpose, x = x (:); endif
  [nr, nc] = size (x);
  if nargin == 1
    n = nr;
  elseif n > nr
    x = [ x ; zeros(n-nr,nc) ];
  elseif n < nr
    x (nr-n+1 : n, :) = [];
  endif

  if n == 1
    w = 1/2;
  else
    w = [ sqrt(1/4/n); sqrt(1/2/n)*exp((-1i*pi/2/n)*[1:n-1]') ] * ones (1, nc);
  endif
  if ( realx && rem (n, 2) == 0 )
    y = fft([ x(1:2:n,:) ; x(n:-2:1,:) ]);
    y = 2 * real( w .* y );
  else
    y = fft ([ x ; flipud (x) ]);
    y = w .* y (1:n, :);
    if (realx) y = real (y); endif
  endif
  if transpose, y = y.'; endif

endfunction

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